An ion cyclotron uses a fixed magnetic field to deflect an ion moving at some velocity through the field. For a spatially uniform magnetic field having a flux density B.sub.0, a moving ion of mass m and charge q will be bent into a circular path in an x-y plane perpendicular to the direction of the magnetic field at an angular frequency .omega..sub.c in accordance with: .omega..sub.c =qB.sub.0 /m. Thus, if the magnetic field strength is known, by measuring the ion cyclotron frequency it is possible in principle to determine the ionic charge-to-mass ratio q/m. In effect, the static magnetic field converts ionic mass into a frequency analog. Because the cyclotron frequencies for singly charged ions (12.ltoreq.m/q.ltoreq.5000) in a magnetic field of about 3 Tesla span a radio frequency range (10 kHz.ltoreq.f.ltoreq.4 MHz), within which frequency can be measured with high precision, the ion cyclotron is potentially capable of offering extremely high mass resolution.
In an ion cyclotron cell, the ions may be formed by irradiation of a neutral gas or solid by various known techniques, including the application of electron, ion, or laser beams directed along the magnetic field. The ions are trapped in the cell because the static magnetic field constrains the ions from escaping anywhere in the x-y plane perpendicular to the field and a small static trapping voltage is applied to the end plates of the cell to prevent the ions from escaping in a z-axis direction parallel to the field. A static electric field is thereby established between the end plates. The application of Gauss's law requires that there be an electric field directed radially outward to balance the "trapping" static electric z field, in order that no net charge be contained in the cell in the absence of ions. The radial electric field opposes the inward-directed Lorentz force EQU F=qv.times.B,
where q is the ionic charge, v is the ion velocity, and B is the magnetic field. The radial electric field thus has the same effect as a decrease in magnetic field strength, thereby decreasing .omega..sub.c, the ICR frequency. Further, if the radial electric field varies non-linearly with radial distance from the center of the cell, then the ICR frequency will vary with the pre-excitation position of the ion in the cell, thereby rendering tune-up more difficult and limiting the ultimate mass resolution available during the excitation event. Even for a perfectly quadrupolar electric field in which the field varies linearly with x-, y-, or z-distance from the center of the cell, the radial electric field acts to limit the highest m/q ion that can be held in the cell.
Fourier transform ICR cells were initially cubic in geometry. Other cell geometries, including orthorhombic, cylindrical, hyperbolic, cells that use multi-annulus trapping plates, and multiple-section cells have been used in attempts to minimize some of the undesirable effects of the trapping potential. Although cubic or orthorhombic cells have the advantage of conceptual simplicity and ease of construction, their electrostatic field description is mathematically complicated. The electric field in such a cell has D.sub.4h symmetry (symmetry upon 90.degree. rotation) and is represented by an infinite series of Laplace's equation. Where the origin of the Cartesian coordinate system is chosen to lie at the center of the cell, expansion of the series in the spatial region near the center of the cell yields an approximately quadrupolar electric potential ##EQU1## where a is the distance between the two end plates, V.sub.T is the trapping voltage applied to the trap voltages, and .alpha. and .gamma. are functions of the cell dimensions.
The equation of motion of an ion of mass m and charge q in a static electromagnetic ion cell is given by ##EQU2## in which the magnetic field direction is assigned to the z-axis, as depicted in FIG. 1, which shows a typical prior art cell.
In the quadrupolar approximation of static electric field, ion motion in the z-direction is separable from motion in the x-y plane in the above equation. There are two eigenfrequencies for x-y motion, the cyclotron and magnetron frequencies, .omega..sub.o and .omega..sub.m, ##EQU3##
The second of the two equations listed immediately above shows that the observed ICR frequency, .omega..sub.o, varies with the trapping potential, V.sub.T. In order to reduce the ICR frequency shift induced by the trapping potential, a rectangular cell elongated along the z-axis was introduced. However, for the elongated cell, the quadrupolar approximation breaks down except near the center of the cell, so that the ICR frequency still varies with pre-excitation position of the ion within the cell.
A cylindrical cell has cylindrical symmetry D.sub..infin.h (symmetry upon infinite rotation) about the z-axis and has the same symmetry as a quadrupolar potential in the radial direction. However, the use of flat trapping end plates leads to a spatially inhomogeneous electrostatic field. The quadrupolar electric potential is again approached only near the center of the cylindrical cell. Therefore, ion cyclotron frequency again varies with ion pre-excitation position in the cell.
The ion cell geometry that most closely approximates the quadrupolar electrostatic potential is the hyperbolic cell. Two trap electrodes are separated by a ring electrode which is cut lengthwise into quadrants. Each of the three electrode surfaces has the shape of a hyperboloid of revolution. The hyperbolic cell produces a near-perfect quadrupolar electric potential within the cell ##EQU4## in which 2r.sub.0 and 2z.sub.0 are the radial and z-dimensions of the cell. The ICR frequency for such a cell is given by ##EQU5## Although ICR frequency is invariant with ion position in a perfect quadrupolar potential, the above equation shows that the ICR frequency still varies with trapping voltage. However, because the actual hyperbolic electrodes are not infinite, an actual hyperbolic cell does not generate a purely quadrupolar electrostatic potential, and the observed ICR frequency still varies somewhat with ICR orbital radius.
"Compensated" trap electrodes have been proposed in attempts to minimize the ICR frequency shift and sidebands resulting mainly from non-zero x-y components of the inhomogeneous electrostatic field divided into annular segments held at different potential. In one design, the segments are coplanar, whereas in a second design the segments are separated along the z-axis for ease of construction. For either cell, the radial component of the electric field is reduced without loss in ion trapping efficiency. In addition, it has been shown that magnetron frequency shift is also reduced by a factor of about 5.
In yet another approach, an elongated three-section cell with a 6:1 aspect ratio has been proposed. If +1 V is applied to each of the two end plates, the electric potential drops to less than 10 .mu.V in the center of such a cell. Thus, radial electric field-induced ICR frequency shifts can be reduced accordingly if ICR detection is limited to the central electric field-free region of the three-section cell. Mass resolution is also improved by a factor of about 2. However, ions nevertheless oscillate back and forth along the z-axis and in fact spend most of their time in the two end sections, in which the electric field again has major radial components, leading to radial loss of high-mass ions. Furthermore, as in the other above-mentioned cell designs, if ions are distributed nonuniformly in the cell before excitation, then ions of different z-amplitude will dephase with respect to each other, leading to inhomogeneous line-broadening.